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which had been given before and which, moreover, is applicable
not solely to mechanics, is that of variational equations, namely,
linear differential equations that determine solutions of a pro-
blem infinitely near to a given solution; the second, that of
integral invariants, which belong entirely to him and play a
capital part in these researches. Further fundamental notions
were added to these, notably those concerning so-called
"periodic" solutions, for which the bodies whose motion is
studied return after a certain time to their initial positions and
original relative velocities.'
The last started a whole department of mathematics, the
investigation of periodic orbits: given a system of planets, or of
stars, say, with a complete specification of the initial positions
and relative velocities of all members of the system at a stated
epoch, it is required to determine under what conditions the
system will return to its initial state at some later epoch, and
hence continue to repeat the cycle of its motions indefinitely.
For example, is the solar system of this recurrent type, or if not,
would it be were it isolated and not subject to perturbations by
external bodies? Needless to say the general problem has not
yet been solved completely.
Much of Poincare's work in his astronomical researches was
qualitative rather than quantitative, as befitted an intuitionist,
and this characteristic led him, as it had Riemann, to the study
of analysis situs. On this he published six famous memoirs
which revolutionized the subject as it existed in his day. The
work on analysis situs in its turn was freely applied to the
mathematics of astronomy.
We have already alluded to Poincare's work on the problem
of rotating fluid bodies - of obvious importance in cosmogony,
one brand of which assumes that the planets were once suffi-
ciently like such bodies to be treated as if they actually were
without patent absurdity. Whether they were or not is of no
importance for the mathematics of the situation, which is of
interest in itself. A few extracts from Poincare^s own summary
will indicate more clearly than any paraphrase the nature of
what he mathematicized about in this difficult subject.
4Let us imagine a [rotating] fluid body contracting by eool-